As noted in the introduction, a quadratic form is \(k\) -isotropic over
\({\mathbb {Q}}\)  if and only if it is \(k\) -isotropic over \({\mathbb {Q}}_p\)  for all primes
\(p\)  and over \(\mathbb {R}\) .
We then apply [1]} with \(U_\infty \)  and \(U_p\)  as defined above, \(S = \emptyset \)  and
\(f,g \in {\mathbb {Z}}[a_{11},a_{12} ,\ldots ,a_{nn}]\)  two distinct
\((n -1) \times (n-1)\)  minors of the generic \(n \times n\)  symmetric
matrix of coefficients.
